I have written a module and in order to run the main code where the module is imported needs to have the language extension {-# LANGUAGE OverloadedStrings #-} or the option needs to be given when things are compiled of course.

If I include the {-# LANGUAGE OverloadedStrings #-} in the module this is no cure and the user will still need to do the same in her code or when compiling.

Is there any way that I can do "something" in the actual module in order to lift this problem from the end user and so to speak "hide" the required language extension?

Note that -XOverloadedStrings is never required. All that extension does is, loosely speaking, translate "I'm a string" to fromString "I'm a string". Without extensions, users can always do the same themselves.